The research is focused primarily on the fundamental issues of high-frequency seismic wave propagation in complex 3-D isotropic and anisotropic structures, which go beyond the traditional approaches. The ray method and its recent extensions as well as its combination with other methods are mainly applied and investigated. The emphasis is put on new, stable, more efficient and flexible algorithms for the forward numerical modelling of seismic wave fields in 3-D inhomogeneous, isotropic and anisotropic structures. Considerable attention is also devoted to applications involving shear waves, converted waves, shear wave splitting and coupling in anisotropic media, particle ground motions, etc.

The research programme started on October 1, 1993. The sixth year of the programme started on October 1, 1998.


October 1, 1998 - September 30, 1999

1. Revisions of packages MODEL, CRT and NET

All future revisions of program packages MODEL, CRT, and NET, delivered to the sponsors in the previous years, will be delivered to the sponsors in the next years of the project.

Packages MODEL and CRT:

Model: General 3-D layered and block isotropic structure, containing isolated bodies, pinchouts, etc. Inside the layers and blocks, the velocity and density varies laterally in all the three dimensions. Dissipation and non-planar topography can be considered.
Type of waves: Arbitrary type of elementary seismic body wave corresponding to the zero-order ray theory (P, S, converted). Arbitrary position of the source.
Computations: Arbitrary position and shape of the source, two-point ray tracing by the shooting method, initial-value ray tracing by numerical integration of ray equations, travel time computation, dynamic ray tracing, paraxial-ray propagator matrix, geometrical spreading, vectorial amplitudes, polarization vectors. The packages may be applied to the evaluation of the elastodynamic ray-theory Green function, and to the computation of the synthetic seismograms. Weak anisotropy and the response of fine layers at the source and receiver sites are under implementation.
Applications: Reflection methods, refraction methods, VSP, hole-to-hole.

Package NET:

Model: General 3-D layered and block isotropic model. The medium parameters are specified at grid points of a 3-D rectangular mesh. The same model as in the complete seismic ray tracing may also be used.
Types of waves: First arrivals, constrained first arrivals.
Computations: Arbitrary position and shape of the source. First-arrival travel times in the whole model are computed. The computations include also travel time of all non-ray waves (such as the first arriving diffracted waves in shadow zones, head waves, etc.). The algorithm of computation is independent on a model complexity.
Applications: Tomography for an arbitrary source-receiver configuration, wavefront reconstruction, etc.

2. Ray tracing and synthetic wavefields in 3-D inhomogeneous anisotropic structures

All future revisions of program package ANRAY, delivered to the sponsors in the previous years, will be delivered to the sponsors in the next years of the project.

Package ANRAY:

Model: 3-D laterally varying structure containing isotropic and anisotropic nonvanishing layers. Specification of elastic parameters inside individual layers either by linear interpolation between isosurfaces of elastic parameters or by a B-spline interpolation within a 3-D rectangular grid.
Types of waves: Arbitrary type of elementary seismic body wave corresponding to the zero-order ray theory (P, S, qP, qS1, qS2, any converted wave).
Computations: Arbitrary position of the point source, numerical integration of ray tracing and dynamic ray tracing equations, calculation of ray vectorial amplitudes, ray Green function (also Green functions in the quasi-isotropic approximation for qS waves), ray synthetic seismograms, particle ground motions.
Applications: Reflection methods, refraction methods, VSP and crosshole configuration.
Planned innovations: a) further development of the quasi-isotropic approximation; b) calculation of KMAH index in anisotropic media; c) checks of stability of anisotropic stiffness tensor in interpolated data; d) introduction of coefficients of conversion at internal interfaces (sea bottom) and at the source; e) tests with ray tracing with RHS specified by polarization vectors; f) study of the possibility of iterative calculation of slowness vectors of waves generated by incidence of a wave at an interface separating anisotropic media; g) further debugging, removing inconsistencies in the description of the package.

3. Sample data for the program packages

The examples of input data for the MODEL package, describing or approximating models delivered by the sponsors or other typical models, will be prepared. Upon request, also the sample data for programs CRT or NET to perform calculations in such models will be prepared. The examples of input data for the ANRAY package, for models delivered by the sponsors or other typical models, will be prepared too.

4. Two-point ray tracing in complex isotropic 3-D structures

The two-point ray tracing code will further be tested and applied to various models. Attention will also be devoted to the calculation of two-point rays diffracted from edges and corner points and to the possibility of calculating the corresponding amplitudes and contributions to the synthetic seismograms.

5. Synthetic seismograms in 3-D isotropic complex structures

Methods to calculate synthetic seismograms in complex structures will be studied, mutually compared and combined.

6. Seismic wave propagation in weakly anisotropic inhomogeneous media

Theoretical investigation of the applicability and accuracy of coupling ray theory, and especially of perturbations from isotropic media. Extensive tests of accuracy with programs based on the quasi-isotropic approximation. Comparison with results of other methods for weakly anisotropic media and for singular regions.

Derivation and programming of formulae for R/T coefficients for weak contrast interfaces separating weakly anisotropic media.

Study of qS waves signatures in arbitrary weakly anisotropic media.

7. Propagation of plane elastic waves in anisotropic 1-D media

Comparison of exact computations with the anisotropic ray-theory computations (with approximately ten terms of the ray series), and with the computations based on the coupling ray theory for weakly anisotropic media. (Anisotropic model with a rotating axis of symmetry.)

8. Reflection and transmission coefficients for inhomogeneous waves in dissipative elastic media

Computation of reflection/transmission coefficients at structural interfaces and thin transition layers between two isotropic dissipative media will be further tested. The main attention will be devoted to frequency-independent reference R/T coefficients.

9. Radiation patterns for point sources situated at the Earth's surface, at structural interfaces, and close to them

Radiation patterns of point sources situated at the Earth's surface, at structural interfaces, at the sea bottom, at thin transition layers, and close to them, will be further studied. Program package RMATRIX by C.J. Thomson, linked to the CRT package, will be used for numerical calculations in 3-D models.

10. Computation of ray-theory travel times at the nodes of 3-D grids

Algorithms of fast calculation of ray-theory travel times in dense rectangular grids will further be investigated. Accuracy and efficiency of the interpolation of ray-theory travel times within ray cells in 3-D models will be further studied and the relevant numerical algorithms will be improved or new ones will be proposed. If possible, attention will also be devoted to the interpolation between different shot and receiver positions.

11. Second-order methods in grid travel-time tracing

Accuracy and efficiency of grid travel-time tracing methods to evaluate first-arrival travel times will further be studied and the possibility of a future extension to 3-D will be considered.

12. Accuracy of seismic modelling

The research will be concentrated mainly on the accuracy of travel-time calculations, on the accuracy of finite-difference modelling of seismic wavefields, and on the accuracy of other modelling methods designed or studied in the framework of the project. The main attention will be devoted to the estimation of the feasibility and costs of ray tracing and to the estimation of the Lyapunov exponents for the macromodels.

13. Seismic tomography

Development of theory and algorithms applicable in seismic travel-time tomography with emphasis on the estimation of their accuracy.

14. Migrations

Resolution and accuracy of migrations will be studied. Attention will be paid to the physical meaning of the migrated sections and to their sensitivity to the macromodel.

15. Finite-difference solutions of elastodynamic equations

sMain attention will be paid to the development of a new 3-D elastic code, based on the 1998 update of the 2-D code with which we were able to solve some previously unstable models.

Hybrid Ray-FD method should be finished. Validation experiments will use comparison with the well tested DW-FD method, where DW is the Discrete Wavenumber method.

A new version of the elastic FD program, interfaced to the MODEL package, will be prepared. The innovation will mainly include the non-planar free surface topography, treated by the vacuum formalism, providing much simpler handling than the other recently available methods. Also included will be a spatially varying Q factor, linearly dependent on frequency. It is also intended to include the option that some blocks are filled with water, but the latter needs further numerical experiments.

16. Textbook on Seismic Ray Theory

An extended and revised version of the textbook on Seismic Ray Theory will be distributed to the consortium members as a special report (approximately 800 pages). The textbook describes the propagation of high-frequency seismic body waves in complex 3-D, laterally varying, isotropic, anisotropic or weakly anisotropic, elastic structures with curved interfaces. The pressure waves in liquid structures are also considered. Arbitrary elementary multiply-reflected waves are discussed, including the converted waves. Ray-theory Green functions are derived and discussed for all possible waves and structures under consideration. The construction of ray-synthetic seismograms and ray-synthetic particle ground motions is described in detail.

17. Concluding remarks

In addition to this programme, we will certainly be responsive to specific technical suggestions and recommendations of sponsors within the general framework of the project. The research in most directions listed above will continue to the next years of the project.

You may download PostScript file (65 kB) with the Research Programme.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .